Algebraic Surfaces

Title page
Prefaces
Chapter 1. Theory and Reduction of Singularities
1. Algebraic varieties and birational transformations
2. Singularities of plane algebraic curves
3. Singulari ties of space algebraic curves
4. Topological classification of singularities
5. Singularities of algebraic surfaces
6. The reduction of singularities of an algebraic surface
Chapter II. Linear Systems of Curves
1. Definitions and general properties
2. On the conditions imposed by infinitely near base points
3. Complete linear systems
4. Addition and subtraction of linear systems
5. The virtual characters of an arbitrary linear system
6. Exceptional curves
7. Invariance of the virtual characters
8. Virtual characteristic series. Virtual curves
Appendix to Chapter II by JOSEPH LIPMAN
Chapter III. Adjoint Systems and the Theory of Invariants
1. Complete linear systems of plane curves
2. Complete linear systems of surfaces in S₃
3. Subadjoint surfaces
4. Subadjoint systems of a given linear system
5. The distributive property of subadjunction
6. Adjoint systems
7. The residue theorem in its projective form
8. The canonical system
9. The pluricanonical systems
Appendix to Chapter III by DAVID MUMFORD
Chapter IV. The Arithmetic Genus and the Generalized Theorem of RIEMANN-ROCH
1. The arithmetic genus P_a
2. The theorem of RIEMANN-ROCH on algebraic surfaces
3. The deficiency of the characteristic series of a complete linear system
4. The elimination of exceptional curves and the characterization of ruled surfaces
Appendix to Chapter IV by DAVID MUMFORD
Chapter V. Continuous Non-linear Systems
1. Definitions and general properties
2. Complete continuous systems and algebraic equivalence
3. The completeness of the characteristic series of a complete continuous system
4. The variety of PICARD
5. Equivalence criteria
6. The theory of the base and the number ρ of PICARD
7. The division group and the invariant σ of SEVERI
8. On the moduli of algebraic surfaces
Appendix to Chapter V by DAVID MUMFORD
Chapter VI. Topological Properties of algebraic Surfaces
1. Terminology and notations
2. An algebraic surface as a manifold M₄
3. algebraic cycles on F and their intersections
4. The representation of F upon a multiple plane
5. The deformation of a variable plane section of F
6. The vanishing cycles δ_i and the invariant cycles
7. The fundamental homologies for the 1-cycles on F
8. The reduction of F to a cell
9. The three-dimensional cycles
10. The two-dimensional cycles
11. The group of torsion
12. Homologies between algebraic cycles and algebraic equivalence. The invariant ρ₀
13. The topological theory of algebraic correspondences
Appendix to Chapter VI by DAVID MUMFORD
Chapter VII. Simple and Double Integrals on an algebraic Surface
1. Classification of integrals
2. Simple integrals of the second kind
3. On the number of independent simple integrals of the first and of the second kind attached to a surface of irregularity q. The fundamental theorem
4. The normal functions of POINCARÉ
5. The existence theorem of LEFSCHETZ-POINCARÉ
6. Reducible integrals. Theorem of POINCARÉ
7. Miscellaneous applications of the existence theorem
8. Double integrals of the first kind. Theorem of HODGE
9. Residues of double integrals and the reduction of the double integrals of the second kind
10. Normal double integrals and the determination of the number of independent double integrals of the second kind
Appendix to Chapter VII by DAVID MUMFORD
Chapter VIII. Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves
1. The problem of existence of algebraic functions of two variables
2. Properties of the fundamental group G
3. The irregularity of cyclic multiple planes
4. Complete continuous systems of plane curves with d nodes
5. Continuous systems of plane algebraic curves with nodes and cusps
Appendix 1 to Chapter VIII by SHREERAM SHANKAR ABHYANKAR
Appendix 2 to Chapter VIII by DAVID MUMFORD
Appendix A. Series of Equivalence
1. Equivalence between sets of points
2. Series of equivalence
3. Invariant series of equivalence
4. Topological and transcendent al properties of senes of equivalence
5. (Added in 2nd edition, by D. MUMFORD)
Appendix B. Correspondences between Algebraic Varieties
1. The fixed point formula of LEFSCHETZ
2. The transcendental equations and the rank of a correspondence
3. The case of two coincident varieties. Correspondences with valence
4. The principle of correspondence of ZEUTHEN-SEVERI
Bibliography
Supplementary Bibliography for Second Edition
Index